148
8 Filtering-Based Robust Fault Detection of Fuzzy Multi-model Jumping System
1 (r ) =
⎡
⎢
⎢
⎢
⎢
⎢
⎢
⎣
11 (r ) ) 12 (r ) Q 13
P(r )A hi (r )
0
0
∗ 22 (r ) Q 23 −P(r )B Fi (r )C hi (r ) 0
0
∗
∗
A 33 (r )
0
0
0
∗
∗
∗
−Q 11
−Q 12 −Q 13
∗
∗
∗
∗
−Q 22 −Q 23
∗
∗
∗
∗
∗ −Q 33
⎤
⎥
⎥
⎥
⎥
⎥
⎥
⎦
,
A 2 (r ) =
⎡
⎢
⎢
⎢
⎢
⎢
⎢
⎣
P(r )B i (r )
A 0108 (r )
A 0109 (r )
A 0110 (r )
0
−P(r )B Fi (r )D di (r ) −P(r )B Fi (r )D f i (r ) ) 0210 (r )
0
0
0
A 0310 (r )
0
0
0
0410 (r )
0
0
0
0
0
0
0
0
⎤
⎥
⎥
⎥
⎥
⎥
⎥
⎦
,
A 3 (r ) =
⎡
⎢
⎢
⎣
−γ
2 I 0
0
0
∗ −γ
2 I 0 A 0810 (r )
∗
∗ −γ
2 I 0910 (r )
∗
∗
∗
−I
⎤
⎥
⎥
⎦ ,
2 (r ) =
⎡
⎢
⎢
⎢
⎢
⎢
⎢
⎣
P(r ))B i (r ) 0 0 0
0
000
0
000
0
000
0
000
0
000
⎤
⎥
⎥
⎥
⎥
⎥
⎥
⎦
,
1 (r ) =
⎡
⎢
⎢
⎢
⎢
⎢
⎢
⎣
P(r ))A i (r ) + A
T
i (r )P(r ) 0 0 P(r ))A hi (r ) 0 0
∗
00
0
00
∗
∗0
0
00
∗
∗∗
0
00
∗
∗∗
∗
0 0
∗
∗∗
∗
∗0
⎤
⎥
⎥
⎥
⎥
⎥
⎥
⎦
,
11 (r ) = P(r )A i (r ) + A
T
i (r )P(r ) +
N
j=1 π i j P(k) + Q 11 ,
12 (r ) =
A i (r ) − A Fi (r ) − B Fi (r )C j (r )
T P(r ) + Q 22 ,
22 (r ) = P(r )A Fi (r ) + A
T
Fi (r )P(r ) +
N
j=1 π i j P(k) + Q 22 .
According to Lemma 8.1, i j (r ) is described as:
i j (r ) = L 11 (r )) i (r, t)L 12 (r ) + L
T
12 (r ))
T
i (r, t)L
T
11 (r )
<λ i L
T
12 (r )L 12 (r ) + λ
−1
i L 11 (r )L
T
11 (r ).
(8.38)
where
L 11 (r ) = col [P(r ) M i (r ) 0 0 0 0 0 0 0 0 0],
L 12 (r ) = [N 1i (r ) 0 0 N hi (r ) 0 0 N 2i (r ) 0 0 0].
Using Schur complements, we have: